Algebraic properties of sup and inf

Let $E,F\subseteq\mathbb{R}$ be nonempty and bounded above/below where needed, and let $c\in\mathbb{R}$.

Order properties:

• If $E\subseteq F$ and both are bounded above, then $sup E\le \sup F$.
• If $E\subseteq F$ and both are bounded below, then $inf E\ge \inf F$.

Translation:

• If $E+c=\{x+c:x\in E\}$, then $\sup(E+c)=\sup E + c,\qquad \inf(E+c)=\inf E + c.$

Scaling:

• If $\lambda\ge 0$ and $\lambda E=\{\lambda x:x\in E\}$, then $\sup(\lambda E)=\lambda\,\sup E,\qquad \inf(\lambda E)=\lambda\,\inf E.$
• If $\lambda<0$, then $\sup(\lambda E)=\lambda\,\inf E,\qquad \inf(\lambda E)=\lambda\,\sup E.$ In particular, $\sup(-E)=-\inf E,\qquad \inf(-E)=-\sup E.$

Finite unions:

• If $E$ and $F$ are bounded above, then $E\cup F$ is bounded above and $\sup(E\cup F)=max \{\sup E,\sup F\}.$ Similarly, if bounded below then $\inf(E\cup F)=min \{\inf E,\inf F\}.$

These rules are used constantly to manipulate bounds and to compare limiting processes.

Proof sketch: All statements follow by checking the defining universal properties of $\sup$ and $\inf$. For example, to show $\sup(E+c)=\sup E+c$, note that $u$ is an upper bound for $E+c$ iff $u-c$ is an upper bound for $E$, then apply least-upper-bound minimality.